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To solve the problem, we need to compute \( A + B^T \), where \( A \) and \( B \) are given matrices. Let’s go through the steps:

### Step 1: Define the Matrices
Matrix \( A \) and \( B \) are defined as:
\[
A = \begin{pmatrix} 7 & -1 & 0 \\ -1 & 7 & -2 \\ 0 & -2 & 7 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & 1 & 2 \\ 1 & -5 & 1 \\ 2 & 1 & -4 \end{pmatrix}
\]

### Step 2: Find the Transpose of \( B \) (\( B^T \))
The transpose of \( B \) is obtained by swapping rows and columns:
\[
B^T = \begin{pmatrix} 3 & 1 & 2 \\ 1 & -5 & 1 \\ 2 & 1 & -4 \end{pmatrix}^T = \begin{pmatrix} 3 & 1 & 2 \\ 1 & -5 & 1 \\ 2 & 1 & -4 \end{pmatrix}
\]

### Step 3: Add \( A \) and \( B^T \)
Now, we perform element-wise addition of \( A \) and \( B^T \):
\[
A + B^T = \begin{pmatrix} 7 & -1 & 0 \\ -1 & 7 & -2 \\ 0 & -2 & 7 \end{pmatrix} + \begin{pmatrix} 3 & 1 & 2 \\ 1 & -5 & 1 \\ 2 & 1 & -4 \end{pmatrix}
\]

### Step 4: Compute the Result
Adding the matrices element-wise:
\[
A + B^T = \begin{pmatrix} 7 + 3 & -1 + 1 & 0 + 2 \\ -1 + 1 & 7 - 5 & -2 + 1 \\ 0 + 2 & -2 + 1 & 7 - 4 \end{pmatrix}
\]
\[
= \begin{pmatrix} 10 & 0 & 2 \\ 0 & 2 & -1 \\ 2 & -1 & 3 \end{pmatrix}
\]

### Final Answer
\[
A + B^T = \begin{pmatrix} 10 & 0 & 2 \\ 0 & 2 & -1 \\ 2 & -1 & 3 \end{pmatrix}
\]

Would you like further details or have any questions?

Here are five related questions to expand your understanding:
1. How do you find the transpose of a matrix, and what properties does it have?
2. What is an element-wise addition of matrices, and when is it defined?
3. How does matrix addition relate to matrix multiplication?
4. Can we generalize this operation for matrices of different sizes?
5. How would \( A - B^T \) differ in computation and result from \( A + B^T \)?

**Tip:** Always check matrix dimensions to ensure they match when performing operations like addition and multiplication.

Discover how to calculate the linear combination A + B^T with matrices A and B. This solution covers matrix addition and transposition, including element-wise operations. Ideal for college-level students studying linear algebra.

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